Astro 6531: Astrophysical Fluid Dynamics
Fall 2024
Instructor:
Profs. Dong Lai & Steve Lantz
- Dong's office: 618 SSB; email: DL57_at_cornell.edu;
Steve's office: 614 SSB or 731 Rhodes; email: steve.lantz_at_cornell.edu
- Office hours: After class on M & W. Other times are fine;
best contact by email, which will usually be answered within 24 hours.
Time & Place:
Monday and Wednesday 10:10 am - 11:25 am, SSB 301
Class website:
https://donglai6.github.io/a6531.html
Description:
Three-credit lecture course, aimed at general astrophysics/physics/engineering
graduate students as well as well-prepared undergraduate students.
A knowledge of fluid dynamics is essential for understanding many of
the most interesting problems in astrophysics (and applied physical
sciences). This course will survey fluid dynamics (including
magnetohydrodynamics and some plasma physics -- time permitting)
important for understanding various astronomical and terrestrial
phenomena. Topics include basic fluid and MHD concepts and equations,
waves and instabilities of various types (e.g., sound, gravity,
Rossby, hydromagnetic, spiral density waves; Rayleigh-Taylor, thermal,
Jeans, rotational, magnetorotational instabilities), shear and viscous
flows, turbulence, shocks and blast waves, etc. These topics will be
discussed in different astrophysical contexts and applications, such
as atmosphere and ocean, star and planet formation, stellar
oscillation/rotation/magnetism, compact objects, interstellar medium,
galaxies and clusters. This course is intended mainly for graduate
students (both theory and observation) interested in astrophysics and
space physics. Students in other areas of applied science and
engineering may find the broad astrophysical and terrestrial
applications useful. Well-prepared undergradate students may also take
the course. No previous exposure to fluid dynamics and astronomy is
required.
The students should be familiar with classical mechanics and
electrodynamics at the intermediate (junior) level, and should be
comfortable with vector calculus (e.g. divergence and curl of a vector).
Organization:
Weekly lectures. There will be about 6-8 problem sets.
No final exam (the last problem set may serve as take-home final exam).
There may be a student project during the second half of the semester (TBD).
Grades will be determined by these HWs, project and participations in class.
Either Letter or S/U grade option is possible. (S = attend lectures and do
70% of HWs with passing grades)
Policy statement: You should abide by the CU Code of Academic
Integrity. You are encouraged to discuss homework with other
students but not to collaborate on writing up the notebook
solutions (and not to copy). Anything that you turn in should be your own
work (homework, project). If you need any academic accommodations
please register with Student Disability Services at the beginning of the
term and bring me the description of the appropriate
accommodations.
Recommended Books:
We will not follow any book too closely, especially when it comes to
astrophysical applications.
- The Physics of Astrophysics II: Gas Dynamics by Frank Shu
You may buy this book, although we will not follow
the book too closely: Some of the material in the book will not be covered,
and some of the material covered in lectures will not be found
in this book. Still it is a good book to have.
- Fluid Mechanics: An Introduction by Michel Rieutord
- Applications of Classical Physics by Kip Thorne and Roger Blandford
- Fluid Mechanics by Landau & Lifshitz
A classic book, good to have. No MHD, but a good presentation of
fluid dynamics, particularly the basics.
- Hydrodynamic and Hydromagnetic Stability by S. Chandrasekhar
Another classic book. Extensive mathematical analyses of fluid
instablilities of all types.
- Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena
by Zeldovich and Raizer
A classic book by a master. Nice discussion of shock waves and
1D flows.
- Plasma Physics for Astrophysics by Russell Kulsrud
We almost certainly will not have time to do most of this book. But
it has nice chapters on MHD.
- Astrophysical Flows by Jim Pringle & Andrew King
A recent textbook from Cambridge Univ. Press
- The Physics of Fluids and Plasmas by Arnab Raichoudhuri
A general textbook, somewhat less comprehensive.
- Astrophysical Fluid Dynamics Lecture Note by Gordon Ogilvie
- Theory of Planetary Atmospheres by Chamberlain and Hunten
Includes both physics and chemistry.
There are many other nice (famous) fluid dynamics books, many containing
interesting applications (not necessarily astrophysical), such as
"Physical Fluid Dynamics" by Tritton
"An Introduction to Fluid Dynamics" by Batchelor
"Waves in Fluids" by Lighthill
"Elementary Fluid Dynamics" by Acheson
"Nonlinear Hydrodynamic Modeling: A Mathematical Introduction" ed. Hampton N. Shirer
(low-order modeling of Rayleigh-Bénard convection)
Fluid Mechanics Films
Topics covered in class:
(suggested reading from Shu, LL=Landau & Lifshitz, TB=Thorne-Blandford,
or other sources).
Here are the topics covered in Spring 2023 . We will cover some of
the topics, and may add some new topics
- 8/26 (DL): Basic assumptions of Fluid Dynamics: collisions in gas
and plasma. Qualitative discussion of collisionless plasma
(gyro-radius condition). Basic fluid equations: mass conservation;
Euler equation, and review of gravity term, magnetic force.
Reading: Shu: Chapter 1; p.45-46; skim Chap.5
(You have learned that in Stellar Structure). LL: P.1-7.
- 8/28 (DL):
Navier-Stokes eqn (just give the viscous force but not deriving
it).
Momentum equation in conservative form; EOS and energy equation.
Barotropic flows (examples of barotropic relations: adiabatic vs
isentropic flows, ISM, degenerate stars, etc). Sound wave (derived).
Reading: LL: sections 1,2,3,7. Shu: p.44-46, p.64, p.107; BT: Skim through
section 13.1-13.6.
- 8/30 (DL): Incompressible flow (why/when valid?).
Properties of inviscid barotropic flow: vorticity equation and
Bernoulli's theorem.
Interpretation of vorticity. Applications: irrotational
flows, coalescing NS binaries, Magnus force.
Reading: Shu: Chap.6. LL: p.8-9, p.12-18.
- 9/4 (DL): Bondi flow (estimate and formal derivation). Bondi-Lyttleton accretion.
Sub-sonic solution (relevance to giant planet formation; qualitative discussion).
Parker stellar wind solution.
Reading: Shu: Chap.6. LL: p.8-9, p.12-18.
- 9/9: No class (makeup class was on 8/30)
- 9/11 (Steve): Viscous flows: 1D equation, viscous stress tensor,
shear and bulk viscosities. Microphysics of viscosity. Scaling and Reynolds number.
Reading: LL: section 15 to Eq. (15.10); section 19 to Eq. (19.2).
- 9/16 (Steve): Videos of flows past a
cylinder and a
sphere at different Re.
Dynamical similarity. Flow passing a sphere: low-Re flow (Stokes flow); behavior as a
function of Re. Viscous boundary layer (estimate), drag force in the presence of BL.
General drag force formula (Epstein, Stokes, etc), application to dust dynamics in
pre-solar nebula.
Reading: Tritton: Chap.3, Chap.8.
- 9/18 (DL): Sound wave generation: oscillating ball, monopolar radiation,
higher-order radiation. Lighthill's law.
Reading: Shu: Chap.6. For those interested in giant planet formation: Section III.C of
Armitage review.
Parker wind.
Sound wave is discussed thoroughly in Chap.8 of LL. See also relevant chapters in Thorne-Blandford
- 9/23 (DL): Sound wave with gravity, Jean's instability. Isothermal
cloud and Jean Mass (digression on Virial theorem). Gravity waves (surface waves on a pond).
Eulerian vs Lagrangian perturbation.
- 9/25 (DL): Gravity waves continued:
Eulerian vs Lagrangian perturbation. Derive the dispersion relation.
Order-of-magnitude discussion on deep-water and shallow-water waves,
wave refraction and Tsunamis.
- 9/30 (DL): Nonlinear shallow water wave equations.
Method of characteristics applied to propagating waves, Burger's equation. Physical discussion of wave steepening.
Rayleigh-Taylor instability (and application to SN explosion).
Reading: Thorne-Blandford: Sections 16.2-16.3
- 10/2 (DL): Waves in plane-parallel atmospheres:
derive local dispersion relation, sound waves vs
gravity waves, Physics of Brunt-Vasala frequency. Convective instability (Schwarzschild criterion).
Reading: Pringle-King, Chap.5.
- 10/7 (Steve): Kelvin-Helmholtz instability: images of jets from
Cen A,
M87,
HH111;
time-lapse animation of Jupiter cloud bands.
Overview of basic methods of analysis used in fluid dynamics.
Derivation of Kelvin-Helmholtz in 2D with gravity stabilization.
Excitation of ocean waves by wind; physical interpretation.
Reading: Chandrasekhar, sections 100-101. LL, section 30.
- 10/9 (Steve): Video explaining
Kelvin-Helmholtz waves
in clouds. Shear flow instability: Rayleigh inflexion point theorem.
Competition between shear and stable stratification: Richardson criterion.
Convective instability including dissipation: Rayleigh-Bénard convection.
Reading: Shu, Chap.8 (pp.93-105). Pringle-King: 10.1-10.3, 10.8. Chandrasekhar: sections 5-12, 15.
- 10/16 (Steve): Images of nonlinear dynamics:
secondary Kelvin-Helmholtz instability
on top of Rayleigh-Taylor, plus
chaotic trajectories
in the Lorenz model and their
sensitivity to initial conditions.
Rayleigh-Bénard convection continued: initial instability and critical Rayleigh number,
Lorenz's low-order model, nonlinear saturation, bifurcations, possibility of chaos.
Reading: Shirer, Chap.2.
- 10/21 (Steve): Convection conclusion: video of
solar granulation.
Boussinesq approximation. Mixing length theory of convection.
Rotation: Rayleigh's criterion for rotational instability.
Fluid dynamics in rotating frame. Rotational distortion.
Reading: Tritton, appendix to Chap.14. Shu, Chap.10.
- 10/23 (Steve): Rotation and Coriolis force: images of
solar internal rotation,
weather maps showing isobars
and wind animation.
Rossby number. Geostrophic flows. Taylor-Proudman theorem.
Rossby waves. Quick overview of global Rossby waves and the CFS instability.
Beta plane approximation.
Reading: Tritton, Chap.16 (2nd ed.). Chamberlain and Hunten, p. 88.
- 10/28 (Steve): Images of p-, g-modes and frequencies in helioseismology from
Berkeley and
Yale.
Kelvin's circulation theorem in a rotating frame. Explanation of Rossby waves based on
theorem. Correspondence of eigenmodes, eigenvalues for local vs. global Rossby waves.
Rotational splitting of resonant mode frequencies in stellar interior (p-, g-modes, etc.).
Helioseismology as a means of inferring differential rotation in the Sun.
Ekman layer intro and estimate of width.
Reading: Chandrasekhar, section 22. Tritton, Chap.16 (2nd ed.).
- 10/30 (Steve): Videos of turbulence developing in a
pipe (Poiseuille flow)
and a planar boundary layer.
Definitions of streamlines, streaklines (dye injection), pathlines (tracer particles).
Ekman spiral. Gravity-inertial waves in rotating fluid (and barotropic inertial waves).
Turbulence: intro, description, definitions.
Reading: Tritton, sections 6.1, 16.5 (2nd ed.). Shu, Chap.9.
- 11/4 (Steve): Video of transition to
turbulence in a wake.
Kolmogorov theory (derive spectrum). 3D and 2D turbulence: conservation laws, energy and
enstrophy cascade (mention Kraichnan theory). Reading: Shu, Chap.9.
- 11/6 (Steve):
Nonlinear sound wave steepening. Shock waves: Shock jump condition. Blast waves:
Sedov-Taylor solution.
Reading: Shu, Chaps.15,17.
- 11/11 (DL): Blast waves: Simple analysis, Sedov-Taylor solution.
Forward an reverse shock, evolution of supernova remnants.
Reading: Shu, Chap.15
- 11/13 (DL): Stellar oscillations: brief observation background.
Stellar oscillations theory: Radial pulsation: equations, estimate
of discrete modes. Radial pulsational instability of massive stars.
- 11/18 (DL): Nonradial pulsations: derive eqns in convenient forms, boundary conditions.
WKB dispersion relation. propagation diagram, mode classification. Gravity waves and
convection: Composition gradients and Ledoux criterion. Double diffusive instabilities,
salt fingers. Nonadiabatic effect and mode excitation (intro): energy equations. Nonadiabatic effect and mode excitation (kappa and epsilon mechanisms).
Application of stellar oscillations.
- 11/20 (DL): Accretion disk intro. Basic equations for axisymmetric thin disks. Boundary layers; origin of viscosity, disk turbulence.
Reading: Shu: Chap.7.
- 11/25 (DL): Waves in disks: dispersion relation, Toomre criterion,
Safraonov-Goldrecih-Ward instability. Spiral density waves: kinematics, wave propgation zone,
Q-barrier, corotation and Lindblad resonances.
Negative energy waves, super-reflection and corotation amplifier.
Reading: Shu, Chap.12,13
- 12/2 (Steve): [TENTATIVE topics through 12/9]
MHD equations: magnetic forces (pressure and tension), magnetic evolution
(diffusion, flux freezing). Hydromagnetic waves.
- 12/4 (Steve):
Magnetostatic equilibrium: pressure-balanced plasma column, sausage and kink
instabilities. Magnetic cloud, Virial theorem. critical mass to flux ratio
(magnetic flux problem of star formation). Ambipolar diffusion. Qualitative
discussion of torsional Alfven waves and magnetic braking. Magneto-Rotational
Instability (MRI): physical discussion.
- 12/9 (Steve):
MRI: mechanical derivation. Magnetic buoyancy (Parker's idea). Magnetic
Rayleigh-Taylor instability. Parker instability.
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